.................................................
......!!!!!!!!...................................
21:28:23 `q001. Note that there are 5 questions in this assignment. Again we have a total of 35 people in a room. Of these, 20 have dark hair and 15 have bright eyes. There are 8 people with dark hair and bright eyes. Let A stand for the collection of people who have dark hair and B for the collection who have bright eyes. The Intersection of these two collections is denoted A ^ B, and stands for the collection of all people who have both dark hair and bright eyes. The Union of these two collections is denoted A U B, and stands for the collection of all people who have at least one of these characteristics. In terms of the diagram you made for the preceding problem, describe the collection A ^ B and the collection A U B. Give the number of people in each of these collections (these numbers are designated by the notation n ( A ^ B) and n(A U B) ). Refer to the diagrams you have made.
......!!!!!!!!...................................
RESPONSE --> A^B has 8 people with dark hair and bright eyes as given to us in the problem. A U B has a total of 27 people. i know this by trial and error but also by the formula A U B= what is in A plus what is in B minus what is in A^B.
.................................................
......!!!!!!!!...................................
21:37:02 `q002. Continuing the preceding example, we let A' stand for the people who are not in the collection A, and we let B' stand for the people who are not in the collection B. What are the characteristics of the people in A', and what characterizes people in B' ? What are n(A ') and n(B '), the numbers of people in A' and B' ?
......!!!!!!!!...................................
RESPONSE --> the characteristics of the people in A' are that they don't have dark hair. there are 15 people in A' because there are 20 people out of 35 in A. 35-20=15. the characteristics of B' are people that don't have bright eyes. there are 20 people that fit this becuase there are 15 people in B out of 35. 35-15=20.
.................................................
......!!!!!!!!...................................
21:40:18 `q003. ( A U B ) ' stands for the everyone outside A U B, and ( A ^ B ) ' stands for everyone outside A ^ B. What characterizes the people in each of these collections, and how many people are there in each?
......!!!!!!!!...................................
RESPONSE --> (A U B)' stands for people with neither dark hair nor bright eyes. there are 35 total people - 27 people in A U B leaving 8 people in (A U B)'. the people in (A^B)' are people that have either one of the characteristics or zero. since only 8 people of 35 have both characteristics which is A^B. then there are 27 people left in (A^B)'.
.................................................
......!!!!!!!!...................................
21:47:56 `q004. How many people are in A ' U B ', and how could those people be characterized? Answer the same for A ' ^ B '.
......!!!!!!!!...................................
RESPONSE --> There are 27 people in A' U B'. Since there are 27 people of 35 in A U B, that leaves 8, and elements cannot be repeated so we had to subject 1 x A^B. these people have either dark hair or bright eyes. there are 8 people in A' ^ B'. Since there are 27 people of 35 in A U B, there are 8 people in the area where A' ^ B'.
.................................................
......!!!!!!!!...................................
21:52:47 `q005. Succinctly describe the relationships between ( A U B ) ', A ' U B ', (A ^ B) ' and A ' ^ B '.
......!!!!!!!!...................................
RESPONSE --> (A U B)' = 8 A' U B' = 27 (A ^ B)'=27 A' ^ B'=8 therefore it is proven that (A U B)'= A'^ B' and A' U B'=(A ^ B)'
.................................................
"